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Practical Abstract Algebra in Swift
.package(url: "https://github.com/typelift/Abstract.git", from: "0.1.0")


A take on abstract algebraic structures, in Swift.

Please note: this is deprecated and no longer maintained. I'm working on a new take on abstract algebra in Swift with swift-abstract.

Abstract is a Swift library that defines protocols for common abstract algebraic structures, along with some concrete implementations for Swift datatypes.

The library also provides tools to test the concrete types for the axioms required by each algebraic structure: tests can then be performed by property-based testing libraries like SwiftCheck.

System Requirements

Abstract supports macOS 10.10+ and iOS 8.3+.


To clone Abstract please run git clone REPOSITORY_URL --recursive to properly clone submodules.


Please add this line to your Package.swift file's dependencies section:

.package(url: "https://github.com/typelift/Abstract.git",
         from: Version(0,0,0))

To use the structures in this library, add "Abstract" to your target's dependencies. To additionally test algebraic laws with the framework, add "Abstract" as a dependency to the relevant testTargets.


Abstract is compatible with Carthage: please refer to Carthage documentation for how to add Abstract as a dependency of your project.

How to use this

Let's see some examples to better understand what Abstract is about. To get an overview of the theory behind all this you can read the rationale.

Cookies appreciation

We're building an app that lets a user request any number of cookies, and then provide a feedback if they're satisfied with the cookies or not. We'd like to track the user interaction with the app in a session object like this:

struct UserSession {
    var lastInteractionDate: Date
    var numberOfCookiesRequested: Int
    var maxCookiesPerRequest: Int
    var averageCookiesPerSession: Double
    var alwaysSatisfied: Bool

At each interaction, we should update the session object.

There are 2 possible interactions:

  • order the cookies;
  • leave a feedback;

We could add 2 methods to UserSession to represent each of those actions:

extension UserSession {
    func orderCookies(_ count: Int) -> UserSession {
        var m = self
        m.lastInteractionDate = Date()
        m.totalCookiesRequested += count
        m.maxCookiesPerRequest = max(m.maxCookiesPerRequest, count)
//        m.averageCookiesPerRequest = ?
        return m
    func leaveFeedback(_ positive: Bool) -> UserSession {
        var m = self
        m.lastInteractionDate = Date()
        m.alwaysSatisfied = m.alwaysSatisfied && positive
        return m

We notice 2 things:

  • each property updates following a specific logic that's not explicitly declared, but must be deduced from the context;
  • there's no way we can update the averageCookiesPerRequest property without keeping track of the total number of requests (something that we don't care about from a business perspective);

We could try and define additional types that explicitly declare how we're supposed to update the various properties.

var lastInteractionDate: Max<Date>
var totalCookiesRequested: Add<Int>
var maxCookiesPerRequest: Max<Int>
var alwaysSatisfied: And

Each property is of a type that specifies how we're supposed to compose two instances: Max<Date> will always keep the highest of two dates, and it's going to be the same for Max<Int> but for numbers; Add<Int> will compose the numbers by adding them, and And will apply the && operation to two Bool. We can then get the wrapped value inside the type with an unwrap function.

Following this strategy we can actually define an Average type that declares a composition function that works as intended:

struct Average {
    private var value: Double
    private var weight: Int
    var unwrap: Double {
        return value
    init(_ value: Double, weight: Int = 1) {
        self.value = value
        self.weight = weight
    static func <> (lhs: Average, rhs: Average) -> Average {
        let sum = lhs.value*Double(lhs.weight) + rhs.value*Double(rhs.weight)
        let count = lhs.weight + rhs.weight
        return Average.init(sum/Double(count), weight: count)

Notice that we're using a <> operator instead of a compose method.

Now we can redefine our UserSession like this:

struct UserSession {
    var lastInteractionDate: Max<Date>
    var totalCookiesRequested: Add<Int>
    var maxCookiesPerRequest: Max<Int>
    var averageCookiesPerRequest: Average
    var alwaysSatisfied: And

extension UserSession {
    func orderCookies(_ count: Int) -> UserSession {
        var m = self
        m.lastInteractionDate = m.lastInteractionDate <> Max(Date())
        m.totalCookiesRequested = m.totalCookiesRequested <> Add(count)
        m.maxCookiesPerRequest = m.maxCookiesPerRequest <> Max(count)
        m.averageCookiesPerRequest = m.averageCookiesPerRequest <> Average(Double(count))
        return m
    func leaveFeedback(_ positive: Bool) -> UserSession {
        var m = self
        m.lastInteractionDate = m.lastInteractionDate <> Max(Date())
        m.alwaysSatisfied = m.alwaysSatisfied <> And(positive)
        return m

This looks cool but boring: thanks to the fact that each of our types composes with <>, we're repeating what's basically the same operation over and over again. It would probably be better to extend the composition operation to UserSession itself, where we compose 2 sessions by composing each pair of properties.

extension UserSession {
    static func <> (lhs: UserSession, rhs: UserSession) -> UserSession {
        return UserSession.init(
            lastInteractionDate: lhs.lastInteractionDate <> rhs.lastInteractionDate,
            totalCookiesRequested: lhs.totalCookiesRequested <> rhs.totalCookiesRequested,
            maxCookiesPerRequest: lhs.maxCookiesPerRequest <> rhs.maxCookiesPerRequest,
            averageCookiesPerRequest: lhs.averageCookiesPerRequest <> rhs.averageCookiesPerRequest,
            alwaysSatisfied: lhs.alwaysSatisfied <> rhs.alwaysSatisfied)

Notice that we're simply merging the properties in pairs: this could actually be defined in a completely generic way, either with a generic Tuple struct where the properties can be combined, or with a code-generation tool like Sourcery.

Now our orderCookies and leaveFeedback methods can actually be redefined as static methods that generate the new session to be combined with the previous one. To do that we need to be able to generate empty values for the properties, that are going to behave to the composition like zero behaves to addition, that is, it leaves the previous value unchanged.

extension UserSession {
    static func orderCookies(_ count: Int) -> UserSession {
        return UserSession.init(
            lastInteractionDate: Max(Date()),
            totalCookiesRequested: Add(count),
            maxCookiesPerRequest: Max(count),
            averageCookiesPerRequest: Average(Double(count)),
            alwaysSatisfied: .empty)
    static func leaveFeedback(_ positive: Bool) -> UserSession {
        return UserSession.init(
            lastInteractionDate: Max(Date()),
            totalCookiesRequested: .empty,
            maxCookiesPerRequest: .empty,
            averageCookiesPerRequest: .empty,
            alwaysSatisfied: And(positive))

Now a bunch of interactions can be easily combined like this:

let finalSession: UserSession = .orderCookies(3)
    <> .orderCookies(5)
    <> .leaveFeedback(true)
    <> .orderCookies(2)
    <> .orderCookies(1)
    <> .orderCookies(4)
    <> .leaveFeedback(true)
    <> .orderCookies(10)
    <> .leaveFeedback(false)
let totalCookiesRequested = finalSession.totalCookiesRequested.unwrap // 25
let maxCookiesPerRequest = finalSession.maxCookiesPerRequest.unwrap // 10
let averageCookiesPerRequest = finalSession.averageCookiesPerRequest.unwrap // 4.167
let alwaysSatisfied = finalSession.alwaysSatisfied.unwrap // false

If we provide an .empty value also for UserSession we can actually collect all the interactions in an Array, and then reduce the collection. This is definitely more convenient and readable, and allows us to separate the collection of the data from their processing. UserSession.empty will naturally be an instance where every property is .empty:

let sessions: [UserSession] = [

let finalSession = sessions.reduce(.empty, <>)

Notice that we could call .reduce(.empty, <>) on any collection were the elements have these two properties:

  • can be composed with <>;
  • have an empty element;

Thus, if we were able to represent these two properties in an abstract way, we could simply define a .concatenated() method for these kinds of collections:

let finalSession = sessions.concatenated()

A type (actually a set, but in programming we really just care about types) equipped with a composition operation that is closed (i.e. non-crashing) and associative, and an .empty value that is neutral to the composition, is usually called a Monoid: all the types defined in this example are monoids, and the Swift type system is strong enough to generically define the interface of a monoid with a protocol. Most of the types and methods used in this example are already defined in Abstract, and to read more about monoids you can refer to the Monoid.swift source file.


FizzBuzz is a classic job interview question used to check the way a candidate approaches the resolution of a problem in code.

The requirement is to write a program that, given a list of integers, prints Fizz for every number divisible by 3, "Buzz" for every number divisible by 5, "FizzBuzz" for every number divisible by both (thus, divisible by 15), and the number itself when it's not divisible by any. It's an easy problem to solve in Swift with a for-in cycle and a couple of if-else statements, but then the interview could proceed by asking the candidate how to make the solution more generic, by removing duplicate logic in the checks for divisibility by 3 and/or 5, and scalable, so that it's easy for example to introduce a "Bazz" word when the number is divisible by 4, that should be combined appropriately with the other 2 words (thus, when the number is divisible by 12, 20 or 60).

This kind of problem can be elegantly solved with some abstract algebra. The fundamental intuition behind a possible abstract algebra approach is that we're dealing with putting things together in various ways.

Let's call "special divisors" the numbers associated to each word (initially, 3 for "Fizz" and 5 for "Buzz"). Every integer could have any number of special divisors, including none, so we're dealing with two kinds of composition:

  • words like "Fizz" and "Buzz" should concatenate when a number has more than one special divisor;
  • the way special words and the number itself concatenate is that the number is ignored when the special word exists, so the latter has priority;

The first composition style is simple concatenation; the second one is a little harder to see as some kind of composition, but it actually is the composition where we get only the first value if it exists (even if both exist), otherwise we get the second, and if none exist we get an "empty" value.

The type representing the string concatenation is simply String, which naturally forms a monoid over concatenation, where the .empty value is just the empty string.

About the simple string concatenation, we'd like to define a function that associates a word to a special divisor: the function will take an Int and return a String, which is going to be "Fizz" or "Buzz". But instead of concatenating words we would actually like to concatenate functions that return words: if we're able to compose the return value, we can actually define a composable function:

func associate(divisor: Int, to text: String) -> Function<Int,String> {
    return Function.init { value in
        value % divisor == 0 ? text : .empty

The Function type is a function type (we get the function back with the .call method) that's also a monoid, so we can compose and concatenate instances of this function like we'd do for String values.

We can easily define our fizz and buzz associations:

let fizz = associate(divisor: 3, to: "Fizz")
let buzz = associate(divisor: 5, to: "Buzz")

Now we can easily generate a function that will transform a number in a word, properly concatenated (like "FizzBuzz" for the number 15), or an empty string if the number has no special divisor.

let transform = [fizz, buzz].concatenated().call

For the second type of composition, Swift already provides a type with the correct semantics; we need to give priority to the first element, but only if it's not .empty, otherwise we yield the second value (.empty or not): that's exactly the composition semantics of Optional, where .empty is .none (or nil) and the composition operation is represented by the ?? operator. Abstract extends Optional with the Monoid protocol, adding the .empty instance and the <> operator. We can define a getWord function that will use Optional<String> to select a value in a composition:

func getWord<T>(for value: T, with association: @escaping (T) -> String) -> String {
	let optionalAssociated = Optional(association(value))
		.flatMap { $0 == .empty ? Optional.empty : Optional($0) }

	let optionalValue = Optional("\(value)")

	return (optionalAssociated <> optionalValue) ?? ""

We can verify the result by putting things together:

let result = (1...100)
    .map { value in
        getWord(for: value, with: transform).unwrap <> "\n"


Now that we've separated the two kinds of composition that are taking place here, we can easily add more words and associations. For example:

let bazz = associate(divisor: 4, to: "Bazz")
let transform = [fizz, buzz, bazz].concatenated().call

This code will add the word "Bazz" to the mix, for all numbers divisible by 4. Notice that in our example, for the number 60 the word "FizzBuzzBazz" will be printed: the order matters here, and we get "Bazz" at the end because we composed our transformation like [fizz, buzz, bazz].

The order matters (not)

Let's assume we have a Process<T> type that encapsulates a run function, executable without any input, that returns a value of type T.

struct Process<T> {
    let run: () -> T

Let's also assume we have a bunch of processes that we want to run, and then combine all the values into a single one. Running all the processes in sequence and then collecting all the values could be tedious and inefficient, but running them in parallel, maybe in a distributed way, could be dangerous, unpredictable and hard to coordinate.

We would like to take advantage of the abstract algebraic structures defined in Abstract to simplify the problem. Everything depends on the T value: it turns out that, if T has certain properties, we can actually run our processes in a distributed and efficient way without any risk.

We have a collection of these processes, and we'd like to run all of them by distributing computation to many units: the processes can require different times to complete, and we'd like to distribute our processing units to different threads/queues.

class ProcessingUnit {
    let context: Context
    init(context: Context) {
        self.context = context
    func execute<T>(_ process: Process<T>, onComplete: @escaping (T) -> ()) {
        context.execute {
            let value = process.run()

The ProcessingUnit uses an execution context on which to run the process: we can represent this with a protocol and make DispatchQueue conform to it:

protocol Context {
    func execute(_ call: @escaping () -> ())

extension DispatchQueue: Context {
    func execute(_ call: @escaping () -> ()) {
        async {

A Collector class will receive all the T values and combine them together: the requirement on T, in this case, is for it to be a monoid, so it has an .empty value and a <> associative composition operation.

class Collector<T: Monoid> {
    private var current: T = .empty
    func append(_ value: T) {
        current = current <> value

Finally, a Distributor class will distribute the work to processing units:

class Distributor {
    let serialContext = DispatchQueue.init(label: "serial")
    func distribute<T>(process: Process<T>, to collector: Collector<T>) {
        ProcessingUnit.init(context: serialContext).execute(process, onComplete: collector.append)

Notice that, if the only constraint that we impose on T is Monoid (in the code above the requirement is implicit) we cannot do much more than distributing the work on a serial queue, because we need the processes to run and complete in the same order as they're passed to the distributor.

An improvement over this would be if T was a CommutativeMonoid: in this case the <> operation is declared to be commutative, which means that the order of composition doesn't matter. This way we can distribute work over a concurrent queue: even if a process ends before one that started earlier, the commutativity will insure that the composition still makes sense.

class Distributor {
    let concurrentContext = DispatchQueue.init(label: "concurrent", attributes: .concurrent)
    func distribute<T>(process: Process<T>, to collector: Collector<T>) where T: CommutativeMonoid {
        ProcessingUnit.init(context: concurrentContext).execute(process, onComplete: collector.append)

To make further improvements we could consider the case of actually distributed executions on stateless, uncoordinated contexts that could fail, restart and execute more than once. If T were a BoundedSemilattice we could actually make this kind of context work anyway because the composition operation, other than commutative, would be declared idempotent, that is, applying it twice would be the same as applying it once.

class Unreliable: Context {
    func execute(_ call: @escaping () -> ()) {
        /// could call more than once

class Distributor {
    let unreliableContext = Unreliable()
    func distribute<T>(process: Process<T>, to collector: Collector<T>) where T: BoundedSemilattice {
        ProcessingUnit.init(context: unreliableContext).execute(process, onComplete: collector.append)

By clearly defining the composition semantics of T we can make assumptions that allow us to be more efficient and less constrained. But making T conform to BoundedSemilattice, CommutativeMonoid or even Monoid will require that the composition operation on T respects some laws like commutativity: Abstract provides functions, defined in the Law namespace, that will allow one to test a type against these laws, thus proving that the type has the required semantics.


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Release Notes

Free Structures and Algebraic Data Types
3 years ago

This release focuses on two main topics: free structures and algebraic data types.

Free structures

4 free algebraic structures were defined as typealiases of existing and brand new data structures, that is:

  • NonEmptyArray (brand new) -> FreeSemigroup;
  • Array -> FreeMonoid;
  • Multiset (a.k.a. Bag, brand new) -> FreeCommutativeMonoid;
  • Set -> FreeBoundedSemilattice.

A free representation for Semiring was also provided, with the type SetM (a Set where elements are monoids).

Algebraic data types

3 algebraic data types were added, that is, data types that exist only for combining other types in interesting ways, that is:

  • Product<A,B>, representing A and B at the same time;
  • Coproduct<A,B>, representing either A or B;
  • Inclusive<A,B>, representing one of the following:
    • only type A;
    • only type B;
    • both types A and B.

These types have definitions for the algebraic structures depending on the contained types: e.g. Product<A,B> is a Monoid if A and B are monoids.

Finally, the Function type was included among the algebraic data types.

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